TITLE: Is this result on tensor products of faithful modules known?
QUESTION [5 upvotes]: On the first page of this preprint, I give a proof (a simplification
of an argument suggested by Ken Goodearl) that if  k  is a field,
A  and  B are k-algebras (not necessarily commutative),  M  is a faithful
left A-module, and  N  is a faithful left B-module, then the left
A ⊗k B-module  M ⊗k N  is again faithful.  It is hard to believe 
this is not classical.  Does anyone know a reference?

REPLY [7 votes]: See Lemma 1.1 of http://www.math.wisc.edu/~passman/balgebra.pdf by Passman. I think he published this years later in Communications in Algebra.